#304784

order	512
nilpotency class	3
isoclinism	isoclinic class of #271643
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,5,8)(3,7,4,6)(9,10)(12,13)(14,15)(17,18,22,19)(20,27,29,32)(21,24,30,25)(23,31,26,28);(1,3,5,4)(2,6,8,7)(10,13)(11,15)(14,16)(17,19,22,18)(20,26,29,23)(21,24,30,25)(27,28,32,31);(1,3,5,4)(2,7,8,6)(9,11,10,14,12,16,13,15)(17,20,21,28,22,29,30,31)(18,23,25,27,19,26,24,32);(1,4,5,3)(2,7,8,6)(9,10,12,13)(11,14,16,15)(17,21,22,30)(18,24,19,25)(20,28,29,31)(23,32,26,27)

Polycyclic presentation:

2^9|a^2=eh,b^2=eh,c^2=ef,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211980);